Problem 39 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ a^{2}(x-a)-b^{2}(x-a) $$

Step-by-Step Solution

Answer

The expression factors to \((x-a)(a-b)(a+b)\).

Step 1: Identify Common Factors

The given expression is \( a^{2}(x-a)-b^{2}(x-a) \). First, observe that the term \((x-a)\) is common in both terms. We can factor \((x-a)\) out of the expression.

Step 2: Factor Out the Common Factor

Factor out \((x-a)\) from the expression: \((x-a)(a^{2} - b^{2})\). Now the expression inside the parentheses is \(a^{2} - b^{2}\), which is a difference of squares.

Step 3: Apply the Difference of Squares Formula

The difference of squares formula is \(a^{2} - b^{2} = (a-b)(a+b)\). Apply this formula to \(a^{2} - b^{2}\): \((a-b)(a+b)\).

Step 4: Write the Final Factored Expression

Combine it all together: \((x-a)(a-b)(a+b)\). This is the fully factored form of the original expression.

Key Concepts

Difference of Squares

The concept of the "difference of squares" appears frequently in algebra. It's a straightforward technique used to simplify certain types of polynomial expressions. The "difference of squares" refers to expressions that fit the form \(a^2 - b^2\). This specific pattern allows us to use a unique factorization formula:

• \(a^2 - b^2 = (a - b)(a + b)\)

Let's break this down:- **Squaring**: To "square" a number means to multiply it by itself. For example, \(a^2\) is \(a\) times \(a\).- **Difference**: The difference between two terms is the result of subtracting one from the other. Here, \(a^2 - b^2\) implies that \(b^2\) is subtracted from \(a^2\).This difference of squares pattern is beneficial because it helps us simplify polynomials into a product of two binomials, making them easier to work with or solve.

Common Factors

Finding common factors is an essential skill in factoring expressions. It involves identifying and factoring out terms that appear in all parts of an expression. In the provided exercise, we started with the expression \(a^{2}(x-a) - b^{2}(x-a)\). Let's look at how we found the common factor here:- **Identify**: Notice that both terms in the original expression share the factor \((x - a)\).- **Factor Out**: Once the common factor \((x-a)\) is identified, it can be "factored out" of the expression.Factoring out common factors simplifies expressions and is an introductory step to applying other factoring techniques. By simplifying this way, it becomes easier to see patterns or apply further strategies, such as the difference of squares explained earlier.

Factoring Expressions

Factoring expressions involves rewriting a mathematical expression as a product of its factors. This technique is essential in simplifying algebraic expressions and solving equations. Let's summarize the key steps to factoring an expression based on the provided exercise:

• **Recognize** the entire expression that needs factoring: \(a^{2}(x-a) - b^{2}(x-a)\).
• **Identify** any common factors in the expression, such as the shared factor \((x-a)\).
• **Factor Out** the common factor, simplifying the expression to: \((x-a)(a^2 - b^2)\).
• **Apply Known Formulas**: Use formulas like the difference of squares to further factor remaining terms. Here, we employ \(a^2 - b^2 = (a - b)(a + b)\) to simplify this result further.

Factoring is a methodical process that builds from simpler steps, allowing you to break down complex expressions. Whether dealing with polynomials, finding roots, or solving equations, mastering these techniques is invaluable.